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In-Depth Information
A monofractal time series can be characterized by a single fractal dimension. In
general, time series have a local fractal exponent
h
that varies over the course of the
trajectory. The function
f
, called the multifractal or singularity spectrum, describes
how the local fractal exponents contribute to such time series. Here the scaling index
h
and its spectral distribution
f
(
h
)
(
)
h
are independent variables, as are the moments
q
τ(
)
and the mass exponent
. The general formalism of Legrendre-transform pairs
interrelates these two sets of variables by the relation, using the sign convention in
Feder [
5
],
q
(
)
=
+
τ(
).
f
h
qh
q
(5.91)
The local Hölder exponent
h
varies with the
q
-dependent mass exponent through the
equality
d
τ(
)
dq
=−
τ
(
q
h
(
q
)
=−
q
),
(5.92)
so the singularity spectrum can be written as
τ
(
f
(
h
(
q
))
=−
q
q
)
+
τ(
q
).
(5.93)
Below we explore these relations using data sets to calculate
τ(
q
)
and its deriva-
τ
(
tive
q
)
.
5.3.3
Some physiologic multifractals
Migraine and cerebral blood flow
Migraine headaches have been the bane of humanity for centuries, afflicting such nota-
bles as Caesar, Pascal, Kant, Beethoven, Chopin and Napoleon. However, their etiology
and pathomechanism have not to date been satisfactorily explained. West
et al
.[
32
]
demonstrate that the characteristics of the time series associated with cerebral blood
flow (CBF) significantly differ between normal healthy individuals and migraineurs,
concluding that migraine is a functional neurological disorder that affects CBF autoreg-
ulation. A healthy human brain is perfused by the laminar flow of blood through the
cerebral vessels providing brain tissue with substrates, such as oxygen and glucose.
The CBF is relatively stable, with typical values in a narrow range even for relatively
large changes in systemic pressure. This phenomenon is known as cerebral autoregu-
lation. Transcranial Doppler ultrasonography enables high-resolution measurement of
middle-cerebral-artery flow velocity (MCAfv) and, even though this technique does not
allow direct determination of CBF values, it does help to elucidate the nature and role
of vascular abnormalities associated with migraine.
Here we examine the signature of migraine pathology in the dynamics of cerebral
autoregulation through the multifractal properties of the human MCAfv time series.
As we mentioned, the properties of monofractals are determined by the global Hurst
exponent, but, as mentioned, there exists a more general class of heterogeneous signals,
multifractals, made up of many interwoven subsets with different local Hurst exponents.
The statistical properties of these subsets may be characterized by the distribution of
